Groups, periodic planes and hyperbolic buildings

无双曲三角建筑中亏格2周期平面的研究

Hyperbolic triangular buildings without periodic planes of genus two

本文研究双曲三角建筑顶点简单传递作用群的曲面子群，借助计算机搜索证明在大多数情况下不存在亏格2周期公寓，为无周期公寓曲面子群的群提供了首批候选。

A 5-chromatic same-distance graph in the hyperbolic plane

The chromatic number of the plane problem asks for the minimum number of colors so that each point of the plane can be assigned a single color with the property that no two points unit-distance apart are identically colored. It is now known that the answer is 5, 6, or 7. Here we consider the problem in the context of the hyperbolic plane. We prove that there exists a distance $d\approx 1.375033509$ so that every 4-coloring of the hyperbolic plane contains two points distance $d$ apart, which are identically colored.

A characterization of the planes meeting a hyperbolic quadric of $\PG(3,q)$ in a conic

In this article, a combinatorial characterization of the family of planes of $\PG(3,q)$ which meet a hyperbolic quadric in an irreducible conic, using their intersection properties with the points and lines of $\PG(3,q)$, is given.

Simple Foundations for the Hyperbolic Plane

H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger's axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed that hyperbolic geometry is incidence geometry. We improve upon Skala's formulation by doing away with Pappus and Desargues altogether, by substituting for them two simpler axioms.

Poisson-Voronoi percolation in the hyperbolic plane with small intensities

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster is asymptotically equal to $πλ/3$ as $λ\to0.$ This answers a question of Benjamini and Schramm.

Hyperbolic plane curves near the non-singular tropical limit

We determine necessary and sufficient conditions for real algebraic curves near the non-singular tropical limit to be hyperbolic with respect to a point, thus generalising Speyer's classification of stable curves near the tropical limit. In order to obtain the conditions, we develop tools of real tropical intersection theory. We introduce the tropical hyperbolicity locus and the signed tropical hyperbolicity locus of a real algebraic curve near the non-singular tropical limit. In the case of honeycombs, we characterise the tropical hyperbolicity locus in terms of the set of twisted edges on th…